Alpha and Beta Formulas — Roots of a Quadratic Equation

For the quadratic equation ax² + bx + c = 0:

1. Sum of Roots (α + β) The sum of the roots is equal to the negative of the coefficient of x divided by the coefficient of x². α + β = −b / a

2. Product of Roots (αβ) The product of the roots is equal to the constant term divided by the coefficient of x². αβ = c / a

If you are given the roots α and β, you can reconstruct the original quadratic equation using this formula:

x² − (Sum of roots)x + (Product of roots) = 0

x² − (α + β)x + αβ = 0

Exam questions often ask for complex expressions involving α and β. You can solve them by rewriting them in terms of (α+β) and (αβ):

1. α² + β² = (α + β)² − 2αβ
2. α³ + β³ = (α + β)³ − 3αβ(α + β)
3. (α − β)² = (α + β)² − 4αβ
4. 1/α + 1/β = (α + β) / αβ
5. α/β + β/α = (α² + β²) / αβ = [(α + β)² − 2αβ] / αβ

Question: If α and β are roots of the equation 2x² − 5x + 3 = 0, find the value of α² + β².

Solution: Here, a = 2, b = −5, c = 3. Step 1: Find sum and product. α + β = −b/a = −(−5)/2 = 5/2 αβ = c/a = 3/2

Step 2: Use the identity. α² + β² = (α + β)² − 2αβ = (5/2)² − 2(3/2) = 25/4 − 3 = 25/4 − 12/4 = 13/4